Some Dynamical Behavior Of Random Dynamical Systems And Nonautonomous Dynamical Systems

This thesis consists of two topics. One topic is about random dynamical systems. The other one concerns about nonautonomous dynamical systems. In the first topic, the topological method and probability method are used to analyze the dynamical behavior of random dynamical systems. A random Conley's theorem for the random dynamical systems on noncompact separate complete space is introduced. In order to consider the applica-tion of Conley index in discrete-time random dynamical systems, a sufficient condition for bifurcation in random dynamical systems is presented. The Conley index for continuous-time random dynamical systems is defined. The almost invariant invariant principles for continuous-time random dynamical systems are obtained. In the second topic, the invariant measure for nonautonomous dynamical systems is defined and considered. Moreover, the state decomposition of nonautonomous dynamical systems is studied.This thesis is organized as follows:In Chapter 1, the origin, developments and main contents of the random dynamical systems and nonautnonomous dynamical systems are recalled. Then the backgrounds and main re-sults of the thesis are summarized.Chapter 2,3,4 and 5 focus on the first topic of the thesis, namely the dynamical behavior of the random dynamical systems.It is known by the Conley's theorem that the chain recurrent set CR(φ) of a deterministic flowφon a compact metric space is the complement of the union of sets B(A)-A, where A varies over the collection of attractors and B(A) is the basin of attraction of A. It has re-cently been shown that a similar decomposition result holds for random dynamical systems on noncompact separable complete metric spaces, but under a so-called absorbing condi-tion. In Chapter 2 a notion of random chain recurrent sets for random dynamical systems is introduced, and then the random Conley's theorem on noncompact separable complete metric spaces without the absorbing condition is proved.In Chapter 3, some properties of random Conley index are obtained and then a suf-ficient condition for the existence of abstract bifurcation points for both discrete-time and continuous-time random dynamical systems is presented. This stochastic bifurcation phe-nomenon is demonstrated by a few examples. In Chapter 4, in order to get the Conley index for continuous-time random dynamical systems, the invariance of isolated invariant set for the continuous-time random dynamical systems is relaxed as the invariance for the integer time, and the essential difficulties for defining the continuous-time Conley index for random dynamical systems is generalized from deterministic flow is resolved. Then the Conley index pair for continuous-time random dynamical systems is introduced. The Conley index of isolated invariant sets for continuous-time random dynamical systems are defined as the connected simple systems.In Chapter 5, an almost sure invariance principle for a class of continuous-time bundle random dynamical systems is obtained. Then the result is applied to derive the random cen-tral limit theorem which is considered by Kifer [Limit thereoms for random transformations and processes in random environments, Trans. Amer. Math. Soc.,350 (1998),1481-1518].Chapter 6 is devoted to the dynamical behavior of nonautonomous dynamical systems. First, invariant measures for nonautonomous dynamical systems are considered. It is shown that there exists an invariant measure for certain asymptotically compact nonautonomous dynamical systems. Recall that decomposition of state spaces into dynamically different components is helpful for understanding dynamics of complex systems. Second, a Conley type decomposition theorem is proved for nonautonomous dynamical systems defined on a non-compact but separable state space. Namely, the state space can be decomposed into a chain recurrent part and a gradient-like part. This result applies to both nonautonomous ordinary differential equations on an Euclidean space (which is only locally compact), and nonautonomous partial differential equations on an infinite dimensional function space (which is not even locally compact).